Realizability conditions for relativistic gases with a non-zero heat flux

TL;DR

This paper derives new realizability conditions for relativistic gases with heat flux, revealing stricter energy and heat flux limits and showing the incompatibility of the Synge equation of state with non-zero heat flux.

Contribution

It introduces a novel set of realizability conditions for relativistic gases with heat flux, extending the Taub inequality and analyzing the limits on adiabatic index and heat flux.

Findings

Minimum energy threshold for non-negative distribution function

Maximum heat flux determined by energy and pressure tensor

Upper limit for adiabatic index decreases with heat flux and pressure anisotropy

Abstract

This work introduces a limitation on the minimum value that can be assumed by the energy of a relativistic gas in the presence of a non-zero heat flux. Such a limitation arises from the non-negativity of the particle distribution function, and is found by solving the Hamburger moment problem. The resulting limitation is seen to recover the Taub inequality in the case of a zero heat flux, but is more strict if a non-zero heat flux is considered. These results imply that, in order for the distribution function to be non-negative, (i) the energy of a gas must be larger than a minimum threshold; (ii) the heat flux, on the other hand, has a maximum value determined by the energy and the pressure tensor; and (iii) there exists an upper limit for the the adiabatic index $\Gamma$ of the relativistic equation of state, and that limit decreases in the presence of a heat flux and pressure anisotropy, asymptoting to a value $\Gamma = 1$. The latter point implies that the Synge equation of state is formally incompatible with a relativistic gas showing a heat flux, except in certain gas states.